Abstract

A novel mathematical model of light scattering by an oriented monodisperse system of triaxial dielectric ellipsoids of complex index of refraction is presented. It is based on an integral equation solution to the scattering of a plane electromagnetic wave by a single triaxial dielectric ellipsoid. Both the position and the orientation of a single representative scatterer in a given coordinate system are considered arbitrary. A Monte Carlo simulation is developed to reproduce the diffraction pattern of a population of aligned ellipsoids. As an example of practical importance, light scattering by a population of erythrocytes subjected to intense shear stress is modeled. Agreement with experimental observations and the anomalous diffraction theory is illustrated. Thus a novel check of the electromagnetic basis of ektacytometry is provided. Furthermore, the versatility of the integral equation method, particularly in the advent of parallel processing systems, is demonstrated.

© 1997 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 8.
  2. G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
    [CrossRef]
  3. G. J. Streekstra, A. G. Hoekstra, R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. 33, 7288–7296 (1994).
    [CrossRef] [PubMed]
  4. N. Uzunoglu, “Theoretical calculations of scattering of electromagnetic waves by precipitation particles,” Ph.D. thesis (University of Essex, Colchester, UK, 1976), pp. 11, 81–141.
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993), Chap. 13.5.
  6. M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
    [CrossRef] [PubMed]
  7. G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
    [CrossRef]
  8. K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
    [PubMed]
  9. M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).
  10. A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
    [CrossRef]
  11. R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 4 (1.2).
  12. J. J. Tuma, Dynamics (Quantum, New York, 1974), Chap. 9.
  13. N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
    [CrossRef]
  14. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11, p. 1574.
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), Chap. 13, p. 429.
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.
  17. A. H. Gandjbakhche, P. Mills, P. Snabre, “Light-scattering technique for the study of orientation and deformation of red blood cells in a concentrated suspension,” Appl. Opt. 33, 1070–1078 (1994).
    [CrossRef] [PubMed]
  18. R. A. Meyer, “Light scattering from red blood cell ghosts: sensitivity of angular dependent structure to membrane thickness and refractive index,” Appl. Opt. 16, 2036–2038 (1977).
    [CrossRef] [PubMed]
  19. A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).
  20. M. Abramowitz, I. A. Sregun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25, pp 887–888.
  21. H. O’Hara, F. J. Smith, “The evaluation of definite integrals by interval subdivision,” Comput. J. 12, 179–182 (1969).
    [CrossRef]

1994

1992

G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
[CrossRef]

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

1990

A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).

1988

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

1987

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[CrossRef] [PubMed]

1981

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[CrossRef]

1978

A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
[CrossRef]

1977

1969

H. O’Hara, F. J. Smith, “The evaluation of definite integrals by interval subdivision,” Comput. J. 12, 179–182 (1969).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Sregun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25, pp 887–888.

Bauersachs, R. M.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

Bayer, R.

G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993), Chap. 13.5.

Breederveld, D.

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[CrossRef] [PubMed]

Evans, B. G.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11, p. 1574.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.

Gandjbakhche, A. H.

Goedhart, P.

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[CrossRef] [PubMed]

Hardeman, M. R.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[CrossRef] [PubMed]

Heethaar, R. M.

Hoekstra, A. G.

Holt, A. R.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
[CrossRef]

Khairullina, A. Y.

A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).

Kinoshita, Y.

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

Kohno, M.

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

Korolevich, A. N.

A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).

Koutsouris, D.

N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
[CrossRef]

Meiselman, H. J.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

Meyer, R. A.

Mills, P.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11, p. 1574.

Murakawa, K.

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

Newton, R.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 4 (1.2).

O’Hara, H.

H. O’Hara, F. J. Smith, “The evaluation of definite integrals by interval subdivision,” Comput. J. 12, 179–182 (1969).
[CrossRef]

Ostuni, D.

G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.

Shubochkin, L. P.

A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).

Smith, F. J.

H. O’Hara, F. J. Smith, “The evaluation of definite integrals by interval subdivision,” Comput. J. 12, 179–182 (1969).
[CrossRef]

Snabre, P.

Sregun, I. A.

M. Abramowitz, I. A. Sregun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25, pp 887–888.

Stamatakos, G.

N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
[CrossRef]

Streekstra, G. J.

Sutera, S. P.

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[CrossRef]

Takeda, T.

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.

Tuma, J. J.

J. J. Tuma, Dynamics (Quantum, New York, 1974), Chap. 9.

Uzunoglu, N.

N. Uzunoglu, “Theoretical calculations of scattering of electromagnetic waves by precipitation particles,” Ph.D. thesis (University of Essex, Colchester, UK, 1976), pp. 11, 81–141.

N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
[CrossRef]

Uzunoglu, N. K.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 8.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), Chap. 13, p. 429.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993), Chap. 13.5.

Wolf, G.

G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
[CrossRef]

Yova, D.

N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
[CrossRef]

Zahalak, G. I.

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[CrossRef]

Appl. Opt.

Biorheology

K. Murakawa, M. Kohno, Y. Kinoshita, T. Takeda, “Application of diffractometry and a linear image sensor to measurement of erythrocyte deformability,” Biorheology 29, 323–335 (1992).
[PubMed]

Clin. Chim. Acta

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[CrossRef] [PubMed]

Clin. Hemorheol.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and myrene aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

Comput. J.

H. O’Hara, F. J. Smith, “The evaluation of definite integrals by interval subdivision,” Comput. J. 12, 179–182 (1969).
[CrossRef]

IEEE Trans. Antennas Propag.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, “An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids,” IEEE Trans. Antennas Propag. AP-26, 706–712 (1978).
[CrossRef]

J. Colloid Interface Sci.

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[CrossRef]

Opt. Eng.

G. Wolf, R. Bayer, D. Ostuni, “Stress-induced rigidification of erythrocytes as determined by laser diffraction and image analysis,” Opt. Eng. 31, 1475–1481 (1992).
[CrossRef]

Opt. Spectrosc. (USSR)

A. N. Korolevich, A. Y. Khairullina, L. P. Shubochkin, “Scattering matrix of a monolayer of optically soft close-packed particles,” Opt. Spectrosc. (USSR) 68, 236–239 (1990).

Other

M. Abramowitz, I. A. Sregun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25, pp 887–888.

N. Uzunoglu, “Theoretical calculations of scattering of electromagnetic waves by precipitation particles,” Ph.D. thesis (University of Essex, Colchester, UK, 1976), pp. 11, 81–141.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993), Chap. 13.5.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 4 (1.2).

J. J. Tuma, Dynamics (Quantum, New York, 1974), Chap. 9.

N. Uzunoglu, G. Stamatakos, D. Koutsouris, D. Yova, “Light scattering by adjacent red blood cells—a mathematical model,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 334–345 (1995).
[CrossRef]

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11, p. 1574.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), Chap. 13, p. 429.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. Cambridge U. Press, Cambridge, UK1992), Chap. 2, pp. 34–41.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 8.

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Figures (7)

Fig. 1
Fig. 1

Irradiation and scattering geometry of an oriented monodisperse system of triaxial dielectric ellipsoids.

Fig. 2
Fig. 2

Coordinate systems used in the single dielectric ellipsoid analysis.

Fig. 3
Fig. 3

(a) Irradiation geometry of a prolate spheroidal erythrocyte with semiaxes ratio c/a = 4 and volume V = 90 µ m3; (b) scattering diagram of Fig. 3(a) if six pivot vectors are used. If seven pivot vectors and larger discretization parameters are used, no noticeable changes appear (convergence has already been established).

Fig. 4
Fig. 4

(a) Scattering geometry for a spherical erythrocyte with volume V = 90 µ m3; (b) comparison of FIEM (dashed curve) with Mie theory (solid curve). Only six pivot vectors were used in FIEM; (c) detail of Fig. 4(b) in the near-forward-scattering interval.

Fig. 5
Fig. 5

(a) Irradiation geometry of a prolate spheroidal erythrocyte with volume V = 90 µ m3 and semiaxes ratio b/a = 4; (b) irradiation geometry of an oriented monodisperse system of prolate spheroidal erythrocytes; (c) diffraction pattern of Fig. 5(b).

Fig. 6
Fig. 6

(a) Irradiation geometry of a prolate spheroidal erythrocyte with volume V = 90 µ m3 and semiaxes ratio b/a = 2; (b) irradiation geometry of an oriented monodisperse system of prolate spheroidal erythrocytes; (c) diffraction pattern of Fig. 6(b).

Fig. 7
Fig. 7

(a) Irradiation geometry of a prolate spheroidal erythrocyte with volume V = 90 µ m3 and semiaxes ratio b/a = 4; (b) irradiation geometry of an oriented monodisperse system of prolate spheroidal erythrocytes; (c) diffraction pattern of Fig. 7(b).

Tables (1)

Tables Icon

Table 1 Convergence of FIEM and Comparison with Mie Theory for a Spherical Erythrocytea

Equations (71)

Equations on this page are rendered with MathJax. Learn more.

E r p = E i r p + 1 - k ˆ n k ˆ n · f k n ,   k i × exp ik 0 r p r p + 0 1 r p 2 ,
E s r = n = 1 N   f k n ,   k i exp ik 0 r - r n r - r n   exp ik 0 x n n = 1 N   f k n ,   k i exp ik 0 r r × exp - i k ˆ n k 0 · r n exp ik 0 x n .
r - r n r - k ˆ n · r n .
x 1 2 a 1 2 + y 1 2 b 1 2 + z 1 2 c 1 2 = 1 .
E r = J i   exp i k i · r + V 1 d r 1 γ G r ,   r 1 · E r 1 ,
γ = k 0 2 4 π   ε - 1 ,
J λ = 1 - k ˆ λ k ˆ λ   for any subscript   λ ,
G r ,   r 1 = 1 + k 0 - 2 G r ,   r 1 .
G r ,   r 1 = exp ik 0 r - r 1 r - r 1 .
r 1 = A 1 · ξ 1 ,
d r 1 = d ξ 1 .
A 1 = cos ψ 1 cos θ 1 sin ψ 1 cos ϑ 1 - sin ϑ 1 - sin ψ 1 cos ω 1 + cos ψ 1 sin θ 1 sin ω 1 cos ψ 1 cos ω 1 + sin ψ 1 sin ϑ 1 sin ω 1 cos ϑ 1 sin ω 1 sin ψ 1 sin ω 1 + cos ψ 1 sin ϑ 1 cos ω 1 - cos ψ 1 sin ω 1 + sin ψ 1 sin ϑ 1 cos ω 1 cos ϑ 1 cos ω 1 .
E r r J i   exp i k i · r + exp ik 0 r r f k s ,   k i + 0 1 r 2 ,
lim r   E r = J i   exp i k i · r + exp ik 0 r r J s · V 1 γ exp - i k s · r 1 E r 1 d r 1 .
f k s ,   k i = J s · V 1 γ exp - i k s · r 1   E r 1 d r 1 ,
f k s ,   k i ,   ê i = f k s ,   k i · ê i .
1 k 0 2   V 1 γ exp - i k 1 · r E r d r = J i 1 k 0 2   V 1 γ   exp k i - k 1 · r d r + 1 k 0 2   V 1 d r γ   exp - i k 1 · r   V 1 d r 1 γ G r ,   r 1 · E r 1 .
E r 1 =   d k 2 C 1 k 2 exp i k 2 · r 1 ,
k 2 = k 2 k ˆ 2 .
  d k 2 K k 1 ,   k 2 · C 1 k 2 = J i U V 1 k 1 ,   k i     for any   k 1 ,
U V 1 k 1 ,   k 2 = 1 k 0 2   V 1 γ exp - i k 1 - k 2 · r 1 d r 1 ,
K k 1 ,   k 2 = 1   U V 1 k 1 ,   k 2 - 1 k 0 2   V 1 d r 1   V 1 d r 1 γ   exp - i k 1 · r 1 × G r 1 ,   r 1 γ   exp i k 2 · r 1 .
exp ik 0 r - r r - r = 1 2 π 2   lim ε 0 +     d p p 2 - k 0 2 - i ε × exp i p · r - r ,
k 0 2 G r ,   r = - 1 4 π δ r - r + × × 1 exp ik 0 r - r r - r ,
× × 1 G r ,   r = - 2 1 G r ,   r ,
K k 1 ,   k 2 = ε 1 U V 1 k 1 ,   k 2 - 1 2 π 2   lim ε 0 +     d p p 2 - k 0 2 - i ε × p 2 1 - p ˆ p ˆ U V 1 k 1 ,   p U V 1 p ,   k 2 .
U V 1 k 1 ,   k 2 = U V 1 K 11 ,   K 21 .
U V 1 K 11 ,   K 21 = 1 k 0 2   V 1 γ exp - i K 11 - K 21 · ξ 1 d ξ 1 ,
K 11 = k 1 · A 1 ,
K 21 = k 2 · A 1 ,
U V 1 k 1 ,   p = U V 1 K 11 ,   P 1 ,
U V 1 K 11 ,   P 1 = 1 k 0 2   V 1 γ   exp - i K 11 - P 1 · ξ 1 d ξ 1
P 1 = p · A 1 .
U V 1 p ,   k 2 = U V 1 P 1 ,   K 21 ,
U V 1 P 1 ,   K 21 = 1 k 0 2   V 1 γ   exp - i P 1 - K 21 · ξ 1 d ξ 1 .
K k 1 ,   k 2 = ε 1 U V 1 K 11 ,   K 21 - 1 2 π 2   lim ε 0 +     d p p 2 p 2 - k 0 2 - i ε × 1 - p ˆ p ˆ U V 1 K 11 ,   P 1 U V 1 P 1 ,   K 21 .
U V 1 K 11 ,   K 21 = a 1 b 1 c 1 ε - 1 j 1 K d 11 - K d 21 c K d 11 - K d 21 c ,
K 11 = k 1 1 - x k 1 2 1 / 2   cos   ϕ k 1 ,   1 - x k 1 2 1 / 2   sin φ k 1 ,   x k 1 · A 1 k 1 H x ,   H y ,   H z ,     k 1 can take complex values ,
x a = cos ϑ a     for any subscript   a ,
K d 11 = k 1 a 1 H x ,   b 1 H y ,   c 1 H z ,
K 21 = k 2 1 - x k 2 2 1 / 2   cos φ k 2 ,   1 - x k 2 2 1 / 2   sin φ k 2 ,   x k 2 · A 1 k 2 Λ x ,   Λ y ,   Λ z ,
K d 21 = k 2 a 1 Λ x ,   b 1 Λ y ,   c 1 Λ z ,     k 2   can take complex values .
k 1 = k 2 = k 0 n 0 .
A - B c = A 2 + B 2 - 2 A · B 1 / 2 ,
A = A Â ,
B = B B ˆ ,
K k 1 ,   k 2 = ε 1 a 1 b 1 c 1 ε - 1 j 1 K d 11 - K d 21 c K d 11 - K d 21 c - 1 2 π 2   a 1 b 1 c 1 2 ε - 1 2   lim ε 0 +     d p p 2 p 2 - k 0 2 - i ε × 1 - p ˆ p ˆ j 1 K d 11 - P d 1 c K d 11 - P d 1 c × j 1 P d 1 - K d 21 c P d 1 - K d 21 c ,
P 1 = p 1 - x p 2 1 / 2   cos   ϕ p ,   1 - x p 2 1 / 2   sin   φ p ,   x p · A 1 p Z x ,   Z y ,   Z z ,
P d 1 = p a 1 Z x ,   b 1 Z y ,   c 1 Z z .
j 1 r 1 - r 2 c r 1 - r 2 c = n = 0 +   2 n + 3 × j n + 1 r 1 c r 1 c   j n + 1 r 2 c r 2 c   T n 1 r ˆ 1 · r ˆ 2 ,
r ˆ i = r i r i c     for   i = 1 ,   2 ,
lim ε 0 +   0 + J ν ap J μ ap   p d p p 2 - k 0 2 - i ε = π i 2   J μ ak 0 H ν 1 ak 0 ,
K k 1 ,   k 2 = 1 ε ε - 1 a 1 b 1 c 1   j 1 K d 11 - K d 21 c K d 11 - K d 21 c - ε - 1 2 2 π 2   a 1 b 1 c 1 2 π ik 0   0 1 d x p   0 2 π d φ p × 1 - p ˆ p ˆ   n = 0 m = 0 n + m = even 2 n + 3 2 m + 3 × j n + 1 K d 11 c K d 11 c   j m + 1 K d 21 c K d 21 c × j m > + 1 k 0 Y h m < + 1 k 0 Y Y 2 × T n 1 P ˆ d 1 · K ˆ d 11 T m 1 P ˆ d 1 · K ˆ d 21 ,
m > = max m ,   n ,     m < = min m ,   n ,
h n x = π 2 x 1 / 2   H n + 1 / 2 1 x     spherical Hankel function ,
Y = a 1 2 Z x 2 + b 1 2 Z y 2 + c 1 2 Z z 2 1 / 2 ,
P ˆ d 1 = 1 Y a 1 Z x ,   b 1 Z y ,   c 1 Z z ,
K d 11 c = n 0 k 0 a 1 2 H x 2 + b 1 2 H y 2 + c 1 2 H z 2 1 / 2 ,
K ˆ d 11 = 1 a 1 2 H x 2 + b 1 2 H y 2 + c 1 2 H z 2 1 / 2 × a 1 H x ,   b 1 H y ,   c 1 H z ,
K d 21 c = n 0 k 0 a 1 2 Λ x 2 + b 1 2 Λ y 2 + c 1 2 Λ z 2 1 / 2 ,
K ˆ d 21 = 1 a 1 2 Λ x 2 + b 1 2 Λ y 2 + c 1 2 Λ z 2 1 / 2 × a 1 Λ x ,   b 1 Λ y ,   c 1 Λ z .
1 - p ˆ p ˆ = 1 - y 2   cos 2 φ p - y 2   cos   φ p   sin   φ p - xy   cos   φ p - y 2   cos φ p   sin   φ p 1 - y 2   sin 2 φ p - xy   sin   φ p - xy   cos   φ p - xy   sin φ p y 2 ,
J i U V 1 k 1 ,   k i = 1 - k ˆ i k ˆ i a 1 b 1 c 1 × ε - 1 j 1 K d 11 - K di 1 c K d 11 - K di 1 c ,
K i 1 = k 0 1 - x k i 2 1 / 2   cos   φ k i ,   1 - x k i 2 1 / 2   sin   φ k i ,   x k i · A 1 k 0 Σ x ,   Σ y ,   Σ z ,
K di 1 = k 0 a 1 Σ x ,   b 1 Σ y ,   c 1 Σ z .
f k s ,   k i = k 0 2 J s ·   U V 1 k s ,   k 2 C 1 k 2 d k 2 .
S = κ j ,   w j | j = 1 ,     ,   n ,
l = 1 n   w l K κ j ,   κ l · C 1 κ l = J i U V 1 κ j ,   k i     j = 1 ,   2 ,     ,   n ,
f k s ,   k i = k 0 2 J s · l = 1 n   w l C 1 κ i × U V 1 k s ,   κ l .
0 1 d x p   integration ,
0 2 π d φ p   integration ,

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